Now suppose Integrating will be In our case we want to find the area under the

Now suppose integrating will be in our case we want

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Now suppose ࠵?(࠵?) = ࠵? : Integrating ࠵?(࠵?) will be ࠵?(࠵?) = - > ࠵? > In our case we want to find the area under the curve between ࠵? = 1 and ࠵? = 2 , so we evaluate ࠵?(1) = - > (1) > = - > ࠵?(2) = - > (2) > = q >
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7 Finally, we subtract F(1) from F(2) to get ࠵?(2) − ࠵?(1) = q > - > = Y > This number is the exact value of the area of the region sketched above. Given the connection with integration, we write this area as: / ࠵? : ࠵?࠵? : - = ࠵?(2) − ࠵?(1) In general the definite integral / ࠵?(࠵?)࠵?࠵? r s denotes the area under the graph of ࠵?(࠵?) between ࠵? = ࠵? and ࠵? = ࠵? as shown in Figure below
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8 The numbers a and b are called the limits of integration, and it is assumed throughout this section that a < b and that ࠵? (࠵?) ≥ 0 as indicated in the Figure above. The technique of evaluating definite integrals is as follows. A function ࠵?(࠵?) is found which differentiates to ࠵?(࠵?) . The new function, ࠵?(࠵?) , is then evaluated at the limits x = a and x = b to get ࠵?(࠵?) and ࠵?(࠵?) . Finally, ࠵?(࠵?) is subtracted from the ࠵?(࠵?) to get the answer. Ie ࠵?(࠵?) − ࠵?(࠵?) The process of evaluating a function at two distinct values of x and subtracting one from the other occurs sufficiently frequently in mathematics to warrant a special notation. We write [࠵?(࠵?)] s r = ࠵?(࠵?) − ࠵?(࠵?) In simple form, the area under the curve f(x) from point a to b is given by the definite integral / ࠵?(࠵?)࠵?࠵? r s = [࠵?(࠵?)] s r = ࠵?(࠵?) − ࠵?(࠵?) Using this notation, the evaluation of / ࠵? : ࠵?࠵? : - = u 1 3 ࠵? > v - : = 1 3 (2) : 1 3 (1) : = 7 3 Note that it is not necessary to include the constant of integration, because it cancels out when we subtract ࠵?(࠵?) from ࠵?(࠵?) .
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9 Example Evaluate the definite integral (a) / 3࠵?࠵? = P : [3࠵?] : P = 3(6) − 3(2) = 12 (b) / (࠵? + 1)࠵?࠵? = : e y ࠵? : 2 + ࠵?z e : = E 2 : 2 + 2F − E 0 : 2 + 0F = 4 Application Consumer Surplus The demand function, ࠵? = ࠵?(࠵?) , sketched in Figure below gives the different prices that consumers are prepared to pay for various quantities of a good. At ࠵? = ࠵? e the price ࠵? = ࠵? e . The total amount of money spent on ࠵? e goods is then ࠵? e ࠵? e , which is given by the area of the rectangle OABC. Now, ࠵? e is the price that consumers are prepared to pay for the last unit that they buy, which is the ࠵? e th good. For quantities up to ࠵? e they would actually be willing to pay the higher price given by the demand curve. The shaded area BCD therefore represents the
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